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A Signal-Based Model of Teleology in Tonal Music

Brand, Mark Andre (2016-03)

Thesis (MEng)--Stellenbosch University, 2016.

Thesis

ENGLISH ABSTRACT: The operationalisation of music’s affective agency is an open problem. This thesis notes a long tradition of associating that agency with goal-directed dynamics in the music, but also that those dynamical forces are consistently described in terms of a theoretical framework which privileges Western tonal music by its reliance on the constructs of pitch and rhythm. Against the backdrop of a revival in interest regarding the question of musical universals, an alternate, geometric intuition is then proposed to account for the privileged status of the perfect cadence in tonal music in terms of a least action principle. An extensive review of harmonic priming experiments underscores the robustness of that status, regardless of a subject’s level of musical education or experience, and strongly suggests a degree of universality. A similarly extensive review of theoretical approaches to sub-Saharan African music reveals how such a geometric model might also account for teleology therein.
The proposed model is operationalised by building a signal processing pipeline containing a recurrent,
complex-valued, gradient-frequency artificial neural network, where each neuron implements non-linear
dynamics according to a fully expanded canonical form describing the behaviour of excitatory / inhibitory
neural populations near a Hopf bifurcation. In the absence of any suitable measure of network energy in the literature, an energy balance relation is constructed whereby to monitor energy flows into and out of the
network. In each of four experiments, the network is first primed with a one second stimulus containing the pitches of a dominant quartad in closed position on G4. Seven targets, being the available diatonic triadic
continuations to the quartad, are then each presented to the network in turn, and the individual components of
the energy balance relation are recorded as the network seeks out its equilibrium.
The first experiment employs sinusoidal stimuli, and wholly excludes inter-neural connectivity. This results in all targets taking the network to the same averaged rate of power supply in each case, being equal to the rate of dissipation at equilibrium, and provides no evidence of an advantage of any single target over the others. The second experiment implements typical “train-and-predict” semantics by allowing non-linear
Hebbian learning to take place during the priming phase, holding learned connections fixed during the
presentation of the target. This produces differentiation between the levels of supplied energy, favouring
targets which contain the pitches of the prime, but does not thereby support our model. The third experiment
allows learning to continue throughout both the priming and target phases, and thereby obtains a lower level
of supplied energy in respect of the tonic target as compared to all others, a result which provides tenuous,
but enticing support for our model. The fourth experiment employed piano tones instead of sinusoidal
stimuli, and produced inconclusive results, thereby suggesting that a more complex network might be
required to deal with real-world audio.